Search

Recent advancements in language technology and Artificial Intelligence have resulted in numerous Language Models being proposed to perform various tasks in the legal domain ranging from predicting judgments to generating summaries. Despite their immense potential, these models have been proven to learn and exhibit societal biases and make unfair predictions. In this study, we explore the ability of Large Language Models (LLMs) to perform legal tasks in the Indian landscape when social factors are involved. We present a novel metric, $\beta$-weighted $\textit{Legal Safety Score ($LSS_{\beta}$)}$, which encapsulates both the fairness and accuracy aspects of the LLM. We assess LLMs' safety by considering its performance in the $\textit{Binary Statutory Reasoning}$ task and its fairness exhibition with respect to various axes of disparities in the Indian society. Task performance and fairness scores of LLaMA and LLaMA--2 models indicate that the proposed $LSS_{\beta}$ metric can effectively determine the readiness of a model for safe usage in the legal sector. We also propose finetuning pipelines, utilising specialised legal datasets, as a potential method to mitigate bias and improve model safety. The finetuning procedures on LLaMA and LLaMA--2 models increase the $LSS_{\beta}$, improving their usability in the Indian legal domain. Our code is publicly released.

Determination of an appropriate warranty length for the lifetime of the product is an important issue to the manufacturer. In this article, optimal warranty length of the product for the combined free replacement and the pro-rata warranty policy is computed based on the Type-II unified hybrid censored data. A non-linear pro-rata warranty policy is proposed in this context. The optimal warranty length is obtained by maximizing an expected utility function. The expectation is taken with respect to the posterior predictive model for the time-to-failure data. It is observed that the non-linear pro-rata warranty policy gives a larger warranty length with maximum profit as compared to linear warranty policy. Finally, a real-data set is analyzed in order to illustrate the advantage of using non-linear pro-rata warranty policy.

This article describes the inferential procedures and Bayesian optimal life-testing issues under Type-II unified hybrid censoring scheme. First, the explicit expressions of expected number of failures, expected duration of testing and Fisher information matrix for the unknown parameters of the underlying lifetime model are derived. Then, using these quantities, the Bayesian optimal life-testing plans are computed in subsequent section. A cost constraint D-optimal optimization problem has been formulated and the corresponding solution algorithm is provided to obtain optimal plans. Computational procedures are illustrated through numerical examples.

\textproc{Weighted Vertex Cover} is a variation of an extensively studied NP-complete problem, \textproc{Vertex Cover}, in which we are given a graph, $G = (V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$ and a parameter $k$. The objective is to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$. In our work, we first study the hardness of \textproc{Weighted Vertex Cover} and then examine this problem under parameterization by $l$ and $k$, where $l$ is the number of vertices with fractional weights. Then, we study the \textproc{Red-Blue Weighted Vertex Cover} problem on trees in detail. In this problem, we are given a tree, $T=(V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$, a function $c:V \rightarrow \{R,B\}$ and two parameters $k$ and $k_R$. We have to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$ and $\sum_{\substack{v \in S\\ c(v) = R}}w(v) \leq k_R$. We tackle this problem by applying different reduction techniques and meaningful parameterizations. We also study some restrictive versions of this problem.

In this paper, we develop double acceptance sampling plan and group acceptance sampling plan for an inverse Weibull distribution based on a truncated life test. We consider the median lifetime of the test units as a quality parameter and obtain the design parameters such as sample size and acceptance number. These plans are obtained under the consumer's risk and the producer's risk simultaneously involved at a certain confidence level. We present a simulation study to support the proposed methods and a comparison between single and double acceptance sampling plans is made. A real data set is also analyzed to illustrate the implementation of the proposed sampling plans. Further, the situation under which the proposed samplings plans can also be used for other percentiles points is discussed. Finally a conclusion is presented., Comment: 14 pages, 8 tables, 1 real data set